Sample design for observing classes of objects

We consider optimal sample designs for observing classes of objects. Suppose we will take a simple random sample of equal-sized sectors from a study population and observe the classes existing on these sectors. The classes might be many different things, for example, herbaceous plant species (in sampling quadrats), microinvertebrate species (in sampling cores), and side effects from a drug (in conducting medical trials). Under a nonparametric mixture model and data from a previous related study, we first estimate the optimal number of sample sectors of a given size. Then for negative binomial dispersions of individuals with a common aggregation parameter k, we consider the optimal size as well as number of sample sectors. A simple test exists to check our common k assumption and our optimal size method requires far less data than would be required by a grid method or other method which utilizes data from sample sectors of several different sizes.